On a Best Choice Problem with Partial Information

Abstract

For a given family $\mathscr{F}$ of continuous cdf's $n$ i.i.d. random variables with cdf $F \in \mathscr{F}$ are observed sequentially with the object of choosing the largest. An upper bound for the greatest asymptotic probability of choosing the largest is $\alpha \doteq .58,$ the optimal asymptotic value when $F$ is known, and a lower bound is $e^{-1},$ the optimal value when the choice is based on ranks. It is known that if $\mathscr{F}$ is the family of all normal distributions a minimax stopping rule gives asymptotic probability $\alpha$ of choosing the largest while if $\mathscr{F}$ is the family of all uniform distributions a minimax rule gives asymptotic value $e^{-1}.$ This note considers a case intermediate to these extremes.